Daisuke Matsushita
We generalize Fujiki relation of Beauville-Bogomolov quadratic form on a projective symplectic variety. As an application, we study a fibre space structure of a projective symplectic variety.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
On the fundamental group of the regular part of Fujiki's compact Kahler symplectic orbifolds
本文计算了Fujiki构造的紧Kahler辛轨道流形正则部分的基本群，并确定了其中的不可约辛轨道流形，回答了A.Perego提出的问题。
Fujiki relations and fibrations of irreducible symplectic varieties
本文针对推广不可约辛流形的奇异复射影簇，证明广义Beauville-Bogomolov形式满足Fujiki关系且秩与光滑情形一致，并用新定义验证其纤维化表现与不可约辛流形相似。
Semipositivity of the orbifold second Chern class in Fujiki's class
研究藤木类中紧致正规解析簇orbifold第二陈类不等式，通过建立切层和余切层一般nef性定理及相关不等式，证明宫冈不等式及orbifold第二陈类半正性。
Fujiki Class $\mathcal C$ Varieties and a Kähler Criterion
本文针对代数几何中广义klt对相关问题，证明了翻转与除子收缩保持凯勒条件，给出藤木类$\boldsymbol{\rm C}$簇的凯勒判别准则，还证明了小$\boldsymbol{\rm Q}$-因子分解和dlt修正的存在性。
Let $Y$ be a smooth hypersurface in a projective irreducible holomorphic symplectic manifold X of dimension 2n. The characteristic foliation $F$ is the kernel of the symplectic form restricted to Y. In this article we prove that a generic leaf of the characteristic foliation is dense in Y if Y has positive Beauville-Bogomolov-Fujiki square.
Beauville--Bogomolov--Fujiki positivity and Moishezonness of complex symplectic manifolds in Fujiki class $\mathscr C$
本文研究Fujiki C类中超Fujiki流形的BBF正性与Moishezon性的关系，证明Moishezon超Fujiki流形的相关性质，给出超Fujiki 4-fold等的对应判据并应用于光滑族的Moishezon轨迹描述。
In 1995, Dan Guan constructed examples of non-Kahler, simply-connected holomorphically symplectic manifolds. An alternative construction, using the Hilbert scheme of Kodaira-Thurston surface, was given by F. Bogomolov. We investigate topology and deformation theory of Bogomolov-Guan manifolds and show that it is similar to that of hyperkahler manifolds. We prove the local Torelli theorem, showing that holomorphically symplectic deformations of BG-manifolds are unobstructed, and the corresponding period map is locally a diffeomorphism. Using the local Torelli theorem, we prove the Fujiki formul…
Periodic points for meromorphic self-maps of Fujiki varieties
本文研究Fujiki类紧复流形上支配亚纯自映射的周期点增长，证明当拓扑次数严格大于其他动力学次数时孤立周期点数量呈指数增长，从而肯定Shou-Wu Zhang的猜想，并在一般情形下给出其指数增长的上界为代数熵。
We study the global geometry of the ten dimensional O'Grady irreducible symplectic variety. We determine its second Betti number, its Beauville form and its Fujiki constant.
We introduce and study the notion of "surface decomposable" variety, and discuss the possibility that any projective hyper-Kähler manifold is surface decomposable, which would produce new evidence for Beauville's weak splitting conjecture. We show that surface decomposability relates to the Beauville-Fujiki relation, a constraint on the cohomology ring of the variety, and that general varieties with $h^{2,0}\not=0$ are not surface decomposable. We also formalize the notion of triangle variety that is useful to produce surface decomposition. We show the existence of these geometric structures o…
We prove several results concerning the intersection cohomology and the perverse filtration associated with a Lagrangian fibration of an irreducible symplectic variety. We first show that the perverse numbers only depend on the deformation equivalence class of the ambient variety. Then we compute the border of the perverse diamond, which further yields a complete description of the intersection cohomology of the Lagrangian base and the invariant cohomology classes of the fibers. Lastly, we identify the perverse and Hodge numbers of intersection cohomology when the irreducible symplectic variet…
Perverse-Hodge complexes for Lagrangian fibrations and symplectic resolutions
该研究针对全纯辛簇的拉格朗日纤维化展开，证明了辛解消情形下反常霍奇复形的辛Hard Lefschetz型定理与对称性，推广了光滑情形的结果并验证了相关猜想，还给出了“反常=霍奇”数值结论的新证明，并应用于奇异Higgs模空间研究。
This paper deals with symplectic varieties which do not have symplectic resolutions. Some moduli spaces of semi-stable torsion-free sheaves on a K3 surface, and symplectic V-manifolds are such varieties. We shall prove local Torelli theorem for symplectic varieties. Some results on symplectic singularities are also included.
Irreducible symplectic varieties via relative Prym varieties
本文利用带反辛对合的 K3 曲面及商曲面线性系构造相对 Prym 簇，给出其为本原或不可约辛簇的判据，并在无穷多维度中产生不可约辛簇序列。
We introduce the notion of a conical symplectic variety, and show that any symplectic resolution of such a variety is isomorphic to the Springer resolution of a nilpotent orbit in a semisimple Lie algebra, composed with a linear projection.
We survey recent results on ample cones and birational contractions of holomorphic symplectic varieties of K3 type, focusing on explicit constructions and concrete examples.
We classify the polarized symplectic automorphisms of Fano varieties of smooth cubic fourfolds (equipped with the Plücker polarization) and study the fixed loci.
We develop the global moduli theory of symplectic varieties in the sense of Beauville. We prove a number of analogs of classical results from the smooth case, including a global Torelli theorem. In particular, this yields a new proof of Verbitsky's global Torelli theorem in the smooth case (assuming $b_2\geq 5$) which does not use the existence of a hyperkähler metric or twistor deformations.
We study derived equivalences of Abelian varieties in terms of their associated symplectic data. For simple Abelian varieties over an algebraically closed field of characteristic zero we prove that the natural correspondence introduced by Orlov, which maps equivalences to symplectic isomorphisms, is surjective.
These notes give an introduction to Geometric Invariant Theory and symplectic reduction, with lots of pictures and simple examples. We describe their applications to moduli of bundles and varieties, and their infinite dimensional analogues in gauge theory and the theory of special metrics on algebraic varieties. Donaldson's "quantisation" link between the infinite and finite dimensional situations is described, as are surprisingly strong connections between the bundle and variety cases.
We prove that the integral Hodge conjecture holds for 1-cycles on irreducible holomorphic symplectic varieties of K3 type and of Generalized Kummer type. As an application, we give a new proof of the integral Hodge conjecture for cubic fourfolds.
We show the termination of any log-minimal model program for a pair $(X,Δ)$ of a symplectic manifold $X$ and an effective $\mathbb R$-divisor $Δ$.
We prove that any ample class on a primitive symplectic variety that is locally trivial deformation of O'Grady's singular 6 dimensional example is proportional to the first Chern class of a uniruled divisor. This result answers a question of Lehn, Mongardi and Pacienza, extending their result for primitive symplectic varieties of this deformation type.
We study families of rational curves on certain irreducible holomorphic symplectic varieties. In particular, we prove that any ample linear system on a projective holomorphic symplectic variety of K3[n]-type contains a uniruled divisor. As an application we provide a generalization of the Beauville-Voisin result on the Chow group of 0-cycles on such varieties.
We give an example of a pair of projective symplectic varieties in arbitrarily large dimensions which are D-equivalent, L-equivalent, and birationally inequivalent.
We determine the Mori cone of holomorphic symplectic varieties deformation equivalent to the punctual Hilbert scheme on a K3 surface. Our description is given in terms of Markman's extended Hodge lattice.
We show that the Andre motive of an irreducible, symplectic variety, which is deformation equivalent to the Hilbert scheme of points on a K3 surface, is an object the category generated by its motive truncated in degree two.
Under some conditions on the deformation type, which we expect to be satisfied for arbitrary irreducible symplectic varieties, we describe which big and nef line bundles on irreducible symplectic varieties have base divisors. In particular, we show that such base divisors are always irreducible and reduced. This is applied to understand the behaviour of divisorial base components of big and nef line bundles under deformations and for K3$^{[n]}$-type and Kum$^n$-type.
We prove that the Kodaira dimension of the n-fold universal family of lattice-polarized holomorphic symplectic varieties with dominant and generically finite period map stabilizes to the moduli number when n is sufficiently large. Then we study the transition of Kodaira dimension explicitly, from negative to nonnegative, for known explicit families of polarized symplectic varieties. In particular, we determine the exact transition point in the cases of Beauville-Donagi and Debarre-Voisin, where the Borcherds Phi_{12} form plays a crucial role.
In this note we introduce and study the almost commuting varieties for the symplectic Lie algebras.
We find all homogeneous symplectic varieties of connected reductive algebraic groups that admit an invariant linear connection.
Varieties admitting a holomorphic symplectic form: LLV algebras and derived equivalences
利用Beauville-Bogomolov分解计算具有全纯辛形式的复光滑射影簇的LLV代数，推广超凯勒簇和阿贝尔簇的已知结果，并证明Orlov猜想对许多此类簇成立，特别对所有四维情形成立。
Non-hyperbolicity of holomorphic symplectic varieties
本文证明了满足有理SYZ猜想的第二贝蒂数至少为5的本原辛簇的非双曲性，并进一步在第二贝蒂数至少为7时建立了Kobayashi伪度量恒为零的结果，从而完成了Kamenova-Lu-Verbitsky的工作。
If X is a symplectic variety emedded in an affine space as a complete intersection of homogeneous polynomials, then X coincides with the nilpotent variety of a semisimple Lie algebra.
Let $X$ be a projective variety of dimension $n\ge1$ over an algebraically closed field of arbitrary characteristic. We prove a Fujiki-Lieberman type theorem on the structure of the automorphism group of $X$. Let $G$ be a group of zero entropy automorphisms of $X$ and $G_0$ the set of elements in $G$ which are isotopic to the identity. We show that after replacing $G$ by a suitable finite-index subgroup, $G/G_0$ is a unipotent group of the derived length at most $n-1$. This result was first proved by Dinh, Oguiso and Zhang for compact Kähler manifolds.
We show that the graded Chow rings of two birational irreducible symplectic varieties are isomorphic. This lifts a result known for the cohomology algebras to the level of Chow rings, despite the non-injectivity the cycle class map. In the special case of general Mukai flops, we present an alternative approach based on explicit calculations.
Irreducible symplectic varieties via K3-del Pezzo double covers
对于2≤n≤10，构造第二Betti数为16≤b₂≤24的2n维不可约辛簇，作为一般K3-del Pezzo双重覆盖上Beauville-Mukai系统的有限辛商群的非平凡终端化。
A singular Moishezon space is not projective even if it is Kaehler. This paper gives a projectivity criterion for Kaehler Moishezon spaces. We shall use this projectivity criterion to show the density of projective symplectic varieties in the Kuranishi family of a (singular) symplectic variety.
Irreducible symplectic manifolds are one of the three building blocks of compact Kähler manifolds with numerically trivial canonical bundle by the Beauville-Bogomolov decomposition theorem. There are several singular analogues of irreducible symplectic manifolds, in particular in the context of compact Kähler orbifolds, and in the context of normal projective varieties with canonical singularities. In this paper we will collect their definitions, analyze their mutual relations and provide a list of known examples.
For simple and simply-connected complex algebraic group G, we conjecture the existence of a functor eta_G from the category of 2-bordisms to the category of holomorphic symplectic varieties with Hamiltonian action, such that gluing of boundaries corresponds to the holomorphic symplectic quotient with respect to the diagonal action of G. We describe various properties of eta_G obtained via string-theoretic analysis. Mathematicians are urged to construct eta_G rigorously.
We shall prove that the Poisson deformation functor of an affine (singular) symplectic variety is unobstructed. As a corollary, we prove the following result. Theorem: For an affine symplectic variety $X$ with a good $C^*$-action (where its natural Poisson structure is positively weighted), the following are equivalent (1) $X$ has a crepant projective resolution. (2) $X$ has a smoothing by a Poisson deformation.
We study symplectic varieties defined over fields of positive characteristics, especially the supersingular ones, generalizing the theory of supersingular K3 surfaces. In this work, we are mainly interested in the following two types of symplectic varieties over an algebraically closed field of positive characteristic, under natural numerical conditions: (1) smooth moduli spaces of sheaves on K3 surfaces and (2) smooth Albanese fibers of moduli spaces of sheaves on abelian surfaces. Several natural definitions of the supersingularity for symplectic varieties are discussed, which are proved to …
We prove the standard conjectures for complex projective varieties that are deformations of the Hilbert scheme of points on a K3 surface. The proof involves Verbitsky's theory of hyperholomorphic sheaves and a study of the cohomology algebra of Hilbert schemes of K3 surfaces.
We show that the moduli spaces of sheaves on a projective K3 surface are irreducible symplectic varieties, and that the same holds for the fibers of the Albanese map of moduli spaces of sheaves on an Abelian surface.
Notes on Poisson deformations of symplectic varieties
本文对先前关于辛流形泊松变形的研究进行了修正和补充。
We shall prove that any small deformation of a Q-factorial projective symplectic variety with terminal singularities is locally rigid; in other words, it preserves the singularity. In particular, many singular symplectic moduli of semi-stable sheaves on K3 have no smoothings via deformations. As an application of the result, we also prove that the smootheness is preserved under a flop in our symplectic case. We conjecture that a projective symplectic variety has a smoothing by a flat deformation if and only if it has a crepant (symplectic) resolution. This conjecture would be true if the minim…
On the finiteness of twists of irreducible symplectic varieties
本文利用不可约辛簇的锥猜想，证明了在特征0的固定有限域扩张下不可约辛簇扭曲的有限性，并讨论了非闭域锥猜想、导出等价扭曲有限性及正特征情形。
The Cone Conjecture for Primitive Symplectic Varieties over a Field of Characteristic Zero and an Application
本文证明了在特征零域上第二贝蒼地数大于五的有理数终端极小原始辛簇的 Kawamata-Morrison 锥猜想，并应用于证明某些条件下纤维丛的相对可动和相对 nef 锥猜想。
We propose a general framework governing the intersection properties of extremal rays of irreducible holomorphic symplectic manifolds under the Beauville-Bogomolov form. Our main thesis is that extremal rays associated to Lagrangian projective subspaces control the behavior of the cone of curves. We explore implications of this philosophy for examples like Hilbert schemes of points on K3 surfaces and generalized Kummer varieties. We also collect evidence supporting our conjectures in specific cases.
A proof of the Kawamata-Morrison Cone Conjecture for holomorphic symplectic varieties of K3^[n] or generalized Kummer deformation type
本文针对K3^[n]型及广义Kummer变形型的射影不可约全纯辛流形，证明了Kawamata–Morrison丰锥猜想的相应版本。
We systematically study the moduli theory of symplectic varieties (in the sense of Beauville) which admit a resolution by an irreducible symplectic manifold. In particular, we prove an analog of Verbitsky's global Torelli theorem for the locally trivial deformations of such varieties. Verbitsky's work on ergodic complex structures replaces twistor lines as the essential global input. In so doing we extend many of the local deformation-theoretic results known in the smooth case to such (not-necessarily-projective) symplectic varieties. We deduce a number of applications to the birational geomet…
We develop criteria for affine varieties to admit uniruled subvarieties of certain dimensions. The measurements are from long exact sequences of versions of symplectic cohomology, which is a Hamiltonian Floer theory for some open symplectic manifolds including affine varieties. Symplectic cohomology is hard to compute, in general. However, certain vanishing and invariance properties of symplectic cohomology can be used to prove that our criteria for finding uniruled subvarieties hold in some cases. We provide applications of the criteria in birational geometry of log pairs in the direction of …
We define the odd symplectic grassmannians and flag manifolds, which are smooth projective varieties equipped with an action of the odd symplectic group and generalizing the usual symplectic grassmannians and flag manifolds. Contrary to the latter, which are the flag manifolds of the symplectic group, the varieties we introduce are not homogeneous. We argue nevertheless that in many respects the odd symplectic grassmannians and flag manifolds behave like homogeneous varieties; in support of this claim, we compute the automorphism group of the odd symplectic grassmannians, and we prove a Borel-…
For any simple, simply connected algebraic group $G$ of type $B,C$ and $D$ and for any maximal parabolic subgroup $P$ of $G$, we provide a criterion for a Richardson variety in $G/P$ to admit semistable points for the action of a maximal torus $T$ with respect to an ample line bundle on $G/P$.
Irreducible symplectic varieties with a large second Betti number
本文证明了非紧Lagrangian fibration存在不可约辛紧化，并应用于证明包含固定cubic fourfold的cubic fivefolds的relative Jacobian fibration可被第二Betti数至少为24的不可约辛簇紧化，属于新形变类型。
We survey recent results about the Torelli question for holomorphic-symplectic varieties. Following are the main topics. A Hodge theoretic Torelli theorem. A study of the subgroup W, of the isometry group of the weight 2 Hodge structure, generated by reflection with respect to exceptional divisors. A description of the birational Kahler cone as a fundamental domain for the W-action on the positive cone. A proof of a weak version of Morrison's movable cone conjecture. A description of the moduli spaces of polarized holomorphic symplectic varieties as monodromy quotients of period domains of typ…
Let $\mbox{IG}(k,2n+1)$ be the odd symplectic Grassmannian. It is a quasi-ho\-mo\-ge\-neous space with homogeneous-like behavior. A very limited description of curve neighborhoods of Schubert varieties in $\mbox{IG}(k,2n+1)$ was used by Mihalcea and the second named author to prove an (equivariant) quantum Chevalley rule. In this paper we give a full description of the irreducible components of curve neighborhoods in terms of the Hecke product of (appropriate) Weyl group elements, $k$-strict partitions, and BC-partitions. The latter set of partitions respect the Bruhat order with inclusions. O…
Independence of the Grothendieck classes of twisted symplectic flag varieties
本文利用非交换动机理论，证明扭曲辛旗簇的格罗滕迪克类在线性及齐次二次意义下，在代数簇的格罗滕迪克环中具有独立性。
We establish P=W and PI=WI conjectures for character varieties with structural group $\mathrm{GL}_n$ and $\mathrm{SL}_n$ which admit a symplectic resolution, i.e. for genus 1 and arbitrary rank, and genus 2 and rank 2. We formulate the P=W conjecture for resolution, and prove it for symplectic resolutions. We exploit the topology of birational and quasi-étale modifications of Dolbeault moduli spaces of Higgs bundles. To this end, we prove auxiliary results of independent interest, like the construction of a relative compactification of the Hodge moduli space for reductive algebraic groups, and…
On the group of zero-cycles of holomorphic symplectic varieties
本研究针对K3曲面上Bridgeland稳定对象的模空间，证明点的周类由对应对象的陈类决定，证实了Junliang Shen等人提出的相关猜想。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。